Prove/Disprove: Inequality for x ≥ 0

[acarchau]

Prove/disprove the following:
For $x \geq 0$, $(1+x)^{1+x} \geq 1+x+x^2$.

 

[adriank]

Well, first of all, do you think it's true?

Second, compare derivatives.

 

[acarchau]

Well, first of all, do you think it's true?

I have a proof which uses the first two derivatives, btw the polynomial consists of the first few three terms of the taylor expansion of $(x+1)^{x+1}$. I was curious if someone could come out with a better proof.

 

[adriank]

Well, I think that's the best way to prove the inequality then.